fm.lautenschlager.confpro.dac 3961.1997

8/11/2019 Fm.lautenschlager.confpro.dac 3961.1997

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1 Copyright 1997 by ASME

Proceedings of DETC97:1997 ASME Design Engineering Technical Conferences

September 14-17, 1997, Sacramento, California

DETC97DAC3961

MULTIOBJECTIVE FLYWHEEL DESIGN:

A DOE-BASED CONCEPT EXPLORATION TASK

Uwe Lautenschlager and Hans A. EschenauerResearch Center for Multidisciplinary Analysis

and Applied Structural Optimization (FOMAAS)University of Siegen, 57068 Siegen, Germany

Email: [email protected]: [email protected]

Farrokh MistreeSystems Realization Laboratory

G.W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlanta, GA 30332-0405, U.S.A.

Email: [email protected]

ABSTRACTIn structural design, expensive function evaluations can be

replaced by accurate function approximations to facilitate theeffective solution of multiobjective problems. In this paper weaddress the question: How can we solve multiobjective shapeoptimization problems effectively using a Design-of-Experiments (DOE) -based approach? To answer this questionwe address issues of creating non-orthogonal experimentaldesigns, when dependencies among the parameters that representshape functions are present. A screening strategy is used to

gain knowledge about the structural behavior within the designspace and the trade-off among multiple design objectives isefficiently modeled through employing response surfaces duringdesign optimizat ion. The shape optimization of a flywheelwhere two conflicting design goals are present is used toillustrate the approach. Our focus is on the method rather thanthe results per se.

KEYWORDS: multicriteria optimization, structuraldesign, design-of-experiments, response surface methodology,compromise Decision Support Problem (DSP)

NOMENCLATUREd i , d i+ deviation variablesh(r) variable disk thicknessm massT kinetic energyP control pointr radiusr, , z cylindrical coordinates material density

Poissons ratio angular velocity stress stress function()i, ()o inner, outer value

()rr, () radial, tangential component()r reference value()adm admissible value()max , () min maximum, minimum value

1 TECHNICAL FOUNDATION

1. 1 Multiobjective OptimizationThe application of Vector, Multiobjective, or Multicriteria

Optimization techniques (MO-techniques) is primarily due tothe fact that nowadays the machine design does not only requirea minimization of costs, but also other objectives likereliability, accuracy, etc. The objectives which are mostlycompetitive and nonlinear do not lead to one solution point forthe optimum but rather to a "functional-efficient" (P ARETOoptimal) solution set, i.e. the decision maker selects the mostefficient compromise solution out of such a set. The use of preference functions transforms the multicriteria optimizationproblem into a scalar substitute problem. This optimizationstrategy is the basic part of the optimization model (Eschenauer,et al., 1990). MO problems can best be solved using numericalmethods based on mathematical programming. For the solutionof optimization problems in structural design, the Three-Column-Model (Eschenauer, et al., 1993) has been appliedsuccessfully in many cases. It consists of the three columns


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structural model, optimization model, and optimizationalgorithms which are integrated within one optimization loop.The general nonlinear MO problem is defined as follows(Eschenauer, et al., 1990):

" Min" xn

f ( x) | h ( x) = 0, g( x) 0{ } , (1)

with n

set of real numbers (n-dimensional), f vector of the k objective functions, x vector of the n design variables, h vector of the q equality constraints, and g vector of the p inequality constraints.

We have a Scalar Optimization Problem (SOP), if only oneobjective is considered. In the case of an MO, objectiveconflicts occur, i.e., no design variable vector allows asimultaneous fulfillment of all objectives. MOs can be solvedthrough defining substitute problems and by that reducing theMO to a SOP like the objective weighting (archimedean), the

constraint-oriented transformation (trade-off method), ormodeling different priority levels (preemptive) for theobjectives. The definition of a substitute problem then reads(Eschenauer, et al., 1990):

{ } Min pn x

f x h x 0 g x 0

= [ ]( ) | ( ) , ( ) , (2)

with the preference or substitute function p[ f ( x)].

1 .2 Compromise Decision Support ProblemModeling MO problems is a key and vital issue in

concurrent design. The initial mathematical problem

description, referred to as the baseline model, can be modeledusing different design strategies with appropriate algorithms andsoftware available in the design domain (Mistree, et al., 1994).In the current case study, the compromise Decision SupportProblem (DSP) is used to model engineering decisionsinvolving multiple trade-offs. The compromise DSP is ahybrid multiobjective programming model (Bascaran, et al.,1987; Mistree, et al., 1993). It incorporates concepts from bothtraditional Mathematical Programming and Goal Programming(GP). The compromise DSP is similar to GP in such a waythat the multiple objectives are formulated as system goals(involving both system and deviation variables) and thedevia tion function is solely a funct ion of the goal deviationvariables. In the compromise formulation, the set of systemconstraints and bounds define the feasible design space and thesets of system goals define the aspiration space . For feasibility,the system constraints and bounds must be satisfied. Asatisficing 1 solution then is that feasible point that achieves the

1 Satisficing solutions are good enough but not necessarily the best

(Simon, 1981).

system goals as far as possible. The solution to this problemrepresents a trade-off between that which is desired (as modeledby the aspiration space) and that which can be achieved (asmodeled by the design space). Two types of objective functionsare permitted. The Archimedean form is used when the relativeimportance of different objectives is known and the Preemptive

form, lexicographic minimum concept (Ignizio, 1985), isapplied to model objectives at different priority levels. Thelexicographic minimum concept is appropriate for use in theearly stages of design where minimal information is availableconcerning the weights of the goals. Different quality scenarioscan quickly be evaluated by changing the priority levels of thegoals to be achieved.

1. 3 DOE-based Concept ExplorationThe formal techniques which support the design and

analysis of experiments are called DOE techniques. DOE is astatistical approach for solving problems from engineering to

social science (Box, et al., 1978; Montgomery, 1991). In theearly stages of designing complex systems, it is advantageousto explore the design space to determine a suitable range of specifications and identify feasible starting points for designoptimization. DOE can be utilized for this purpose. Based onDOE techniques, response function approximations arefrequently used to replace expensive function evaluations instructural optimization (Barthelemy and Haftka, 1993; Roux, etal., 1996; Schoofs, et al., 1992). However, the problems areoften limited in complexity. Our frame of reference for themultiobjective design of a flywheel is a Robust Concept

Exploration Method (RCEM) where several statistical methods,i.e., DOE techniques, response surface methodology and robustdesign are integrated within the compromise DSP (Chen, et al.,1996). The RCEM is developed as a step-by-step approach tofacilitate quick evaluation of different design alternatives and thegeneration of a ranged set top-level design specifications withquality considerations (robust design) in the early stages of design. It is primarily useful for designing complex systemsand computationally expensive design analysis. There are fourmajor steps involved, namely, "classify design parameters","screening experiments", "elaborate the response surfacemodels" and "generate top-level design specifications withquality considerations". In this paper, we are considering thefollowing steps in our design approach:

Step 1: DOE-based screening experiments and designmodel reduction (Section 5.1),

Step 2: secondary experiments and response surface models(Section 5.2),

Step 3: top-level design specifications through modelingthe design trade-off (Section 5.2).


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In Section 2 we develop the structural analysis model of aflywheel that is used to illustrate our approach. Theformulation of the compromise DSP for the flywheel is givenin Sect ion 3. Experimental designs for shape optimizationproblems are introduced in Section 4 and in Section 5, ourapproach is applied to the flywheel design and solutions are

given.

2 STRUCTURAL MODELING AND ANALYSISThe classical design objective for a flywheel is to maximize

the capacity of the stored kinetic energy while satisfyingconstraints on the mass and stress limits. This problem hasbeen studied extensively in the design literature, e.g., (Bergerand Porat, 1988; Bhavikatti and Ramakrishnan, 1980; Sandgrenand Ragsdell, 1983). Recent interest has been influenced by theavailability of affordable high strength, lightweight compositematerials, advancements in low loss bearings, and demands forvehicular and electric utility energy storage (Grudkowski, et al.,

1996). The emphasis in the current study is to examine theapplicability of a DOE-based and response surface approach in ashape optimization problem with multiple objectives. Thenon-uniform flywheel profile has been represented in form of various F OURIER series. Sandgren (Sandgren and Ragsdell,1983) consider the shape to be described through a sine serieswith one to seven coefficients. Vadde (Vadde, 1995) uses aFOURIER series with three equal coefficients for the sine andcosine terms. Rao and coworkers (Rao, et al., 1996) includethree sine and three cosine terms in their work. In this study,the nonuniform profile of the flywheel is represented throughBEZIER curves. A sketch of the profile is given in Figure 1.

disk

shaft

h (

r )

r or i

r

z

Figure 1 -- Sketch of a flywheel profil e

The structural analysis of the flywheel can be separated intothe analysis of the shaft and the analysis of the disk. The torqueof shaft can be transmitted into the disk with a key or a shrink fit. The shaft and disk are both considered rigid and the systemis assumed to be stable without whirling. For the currentmodel the role of the shaft is neglected for simplicity, althoughthe shaft-disk connection is a very important aspect for the

boundary condition during disk analysis. We analyze the stressstate, the mass and the kinetic energy. The mathematical modelfor calculating the stress distribution in the disk withnonuniform profi le is adopted from (Sandgren and Ragsdell,1983). Shear stresses are neglected, therefore, the radial and

tangential stresses, rr and respectively, are the principalstresses and assumed to be uniform across the thickness. Thefollowing equation of equilibrium

( )d dr

hr h hr rr + =2 2 0 (3

is obtained from applying a force balance on a small disk element, where h = h(r) is the variable thickness at radius r, ithe material density, and is the angular velocity. A stressfunction := hr rr is defined such that Eq. (3) becomes

h =ddr

+ 2 hr2 . (4

Applying the material law equations and using the fact of radial symmetry, this equation can be simplified into a second-order differential equation in the stress function :

r 2d 2 dr 2

+ r ddr

+(3 + )2hr 3 rh

dhdr

(rddr

) = 0

(5

where is the Poissons ratio. Due to the non-constantthickness h(r), Eq. (5) cannot be solved analytically. A fourthorder R UNGE -KUTTA method is used to solve this differentialequation numerically. Two boundary conditions for the stressfunction have to be specified. They result from thesimplifying assumption that the radial stress rr is zero at theinner and outer radii and therefore is also zero at both radii.All the necessary points for the calculation have to be derivedfrom the B EZIER curve definition. Using a shooting method,the solution is obtained when both boundary conditions aresatisfied. Given a numerical solution of this equation, rr and are calculated as

rr =hr

, (6a

=1h

ddr

+2 r2 (6b

respectively. If the stresses of the disk exceed an admissiblevalue, failure can be expected. As a failure criterion, thereference stress r is used which is determined by the shearstrain energy hypothesis according to VON M ISES . Once theradial and tangential stress components are obtained, thereference stress r is calculated from

r =12

rr ( )2 + rr2 +2( )

12

, (7


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which has to be smaller than the admissible stress adm . Anexpression for the mass and the kinetic energy of the disk can bederived in a simple way. The disk mass m is calculated from

m hrdr r

r

i

o

= 2 (8)

and the kinetic energy T from

T hr dr r

r

i

o

= 2 3 (9)

Both integrals are solved numerically using a S IMPSON srule, where the required points have to be derived from theBEZIER curve definition. The analysis model is implementedusing the before mentioned numerical methods for solving thedifferential equation and the integrals. The implementation of this model is verified through a comparison of radial andtangential stresses with the analytically derived solution for ahollow rotating disk of constant thickness.

3 PROBLEM FORMULATIONThe objective for the flywheel design is to store as much

kinetic energy as possible for a given angular velocity, whilelimits on the allowable stress, the mass, and geometric boundshave to be satisfied. However, a designers objectives may alsoinclude a low system mass or low stresses. An increase inmass will result in an increase in kinetic energy. Here, wemodel the conflict between achieving low stress and high energystorage, which represents a vector optimization problem. Theflywheel design is a shape optimization problem that can besolved with MO-techniques.

Shape optimization is a sub-task of structural optimizationwhich deals with the optimal layout of structures. In contrastto parameter optimization problems, where we search foroptimal design variable values or parameter configurations, insolving shape optimization problems, we search for optimal

functions that describe the shape of a structure (Eschenauer, etal. , 1994). For practical solutions to shape optimizationproblems, so-called direct methods transform the original shapeoptimization problem into a parameter optimization problem,usually through the introduction of special shape functions.The choice of the shape functions is extremely important,because the original solution space will be reduced and theoptimization result will become influenced, too. Highflexibility with a small number of free parameters to describesurfaces or lines is preferred to render a large variety of possibleshapes during optimization. Among the most practical shapefunctions are straight line segments, C 1-connected coupled line-arc curves, B-spline, and B EZIER curves. A B EZIER curve, asused in this study, is controlled in a predictable way bychanging only a few simple parameters, the control point

coordinates Each control point influences the whole curve.This is called a global control property. Control points areusually not lying on the curve, except the first and last one.The tangent at these points is determined through the second andsecond last points (Mortenson, 1985; Weinert, 1994). In orderto solve a multicriteria shape optimization problem, it is first

transformed into an MO problem by means of a direct shapeoptimization strategy. By applying a preference function, theMO is transformed into a SOP. Here, this problem is solvedusing a compromise DSP formulation.

The structural quantities to be varied are called designvariables. Here, they are geometrical dimensions of the disk andcontrol points of the shape function. Four control points areused to define the disk profile (Figure 2). Each of the controlpoints has two degrees of freedom (r- and z-components) whichresults in eight possible design variables. While the disk dimensions, the inner and outer disk thickness (h i, h o) and radii(ri, r o), define the coordinates of the two curve end points, thetwo intermediate control point coordinates define the curvatureof the profile.

r or i

r

z

h o

h i

P1

P2 P3

P4

Figure 2 -- Profile definition with B EZIER curves

For the structural analysis and optimization, the used modelparameters include material, load, constraint and goalparameters, which are specified in Table 1.

Table 1 -- Model p arameters

Parameter Valuematerial density 7.85 10 -6 kg/mm 3

Poissons ratio 0. 3angular velocity 630.0 s -1

admissible stress adm 250.0 MPamax. thickness hmax 100.0 mmmin. thickness hmin 20.0 mm

max. mass mmax 100.0 kgkinetic energy target GT 1500.0 kJ

stress target G 100.0 - 250.0 MPa


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The trade-off in this design problem is modeled as acompromise DSP. Both objectives are modeled in a preemptiveformulation, i.e., on different priority levels. The priority of these goals can then be easily switched depending on adesigners preference. Each goal has two associated deviationvariables d i

and d i+ which indicate the extent of deviation from

the target. The product constraint d i di+ =0 ensures that atleast one of the deviation variables for a particular goal will bezero. If the problem is solved using a vertex solution scheme(as in the ALP-algorithm (Mistree, et al., 1993)) this conditionis automatically satisfied. The standard flywheel designproblem can now be formulated as follows:

Given The analysis model of the flywheel; assumptions used to

model the domain of interest. System parameters: , , , max , hmax , hmin , mmax , mmin ,

GT, GFind

System Variables : ri, r o , h i, h o , P 2z , P 3z , P 2r, P 3r Deviation Variables : Kinetic Energy: d 1

,d 1+

Total Stress: d 2 ,d 2

+

Satisfy System Constraints

Maximum disk stress: maxri r ro

( r ) maxMaximum disk thickness: max

ri r ro(h) h max

Minimum disk thickness: minri rro

(h) h minMaximum disk mass: m mmax

System Goals

High goal for kinetic energy: TG T

+d 1 d1+ = 1

Low goal for total stress: max( t )G

+d 2 d2+ = 1

Bounds20mm ri 50mm 0mm P2z ,P 3z 40mm100mm ro 300mm 1 P2r* ,P3r* 120mm h i , h o 100mm

d idi

+ =0 , with d i, d i+ 0Minimize

Deviation Function (Preemptive, using lexicographicminimum concept 2):Z = [f 1(d 1

),f 2 (d 2, d 2

+) ]

2 Given an ordered array f (i) = (f 1 ,f 2 , ..., f n)

(i) of non-negative elements f k s,

the solution given by f (1 ) is preferred to f (2) if f k (1) < f k

(2) and f i(1) < f i

(2 ) for

i = 1, ... , k-1; that means all higher-order elements are equal. If no other

solution is preferred to f (1) , then f (1) is the lexicographic minimum.

We introduce the relative, radial control point componentsP2r

* and P 3r* , which are dependent on the values of r i and r o and

therefore have normalized bounds. These values are thenconverted into the actual parameter values, P 2r and P 3r. Anotherapproach is to restrict their position through introducingadditional constraints.

Before the compromise DSP is solved, our approachinvolves a design problem reduction and functionapproximations with response surface models. The requiredexperimental designs for this procedure are introduced in thefollowing section.

4 EXPERIMENTS FOR SHAPE PROBLEMSBy using design of experiments, it is possible to study the

effects of parameters on the system responses. A model thatdescribes the data is assumed and the significance of each of theparameter estimates can be determined through statisticalanalysis after conducting a series of experiments. There exists atrade-off between the number of experiments and the accuracy of the estimated model or approximation. The choice of experimental designs is up to the designer dealing with theproblem and his specific needs, time and economic constraints.When designing complex systems, the number of designvariables can be large in the early design stages, when thedesign freedom is still high and the design knowledge is low.To improve the experimentation efficiency, a sequential strategyof performing low order screening experiments and buildinghigher order response surface models based on secondaryexperiments over a reduced design space is employed. Afterscreening, the number of variables might be reduced to the mostcritical ones and a region of interest might be identified withinthe whole design space.

There exists a large variety of experimental designs, e.g.,Full and Fractional Factorial Designs, Orthogonal Arrays (OA),or Central Composite Designs (CCD) (Box, et al., 1978;Montgomery, 1991). In a full factorial design, all possiblecombinations m k of factor levels are investigated, where k factors are selected at m levels. The number of experimentsincreases exponentially with the number of factors and istherefore not always appropriate. It is possible to estimate allmain and all interaction effects. To reduce the number of experiments to a more practical level, fractional factorial designsof the form m k-p have been introduced, where the generator p

describes the fraction of the full design. It is assumed thatcertain higher order interactions are negligible. As a result it isnot possible to estimate all effects separately, some effects willbe confounded. Taguchi proposed orthogonal arrays to theengineering community. OAs are the smallest fractionalfactorial design to study the main factor effects. Orthogonalarrays are denoted by L x , where x represents the number of experiments (Phadke, 1989). It is assumed that factorinteractions do not exist, therefore main effects can be


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confounded with two-factor interactions, which is not alwaysdesi rable . Some of the columns in an experimental designmatrix can be used to study interactions, however, a designerdoes not necessarily know which interactions may exist in thesystem.

In our sequential experimentation strategy, two-level

screening experiments are used to fit a linear model of the form

y = b 0 + bii =1

k

x i . (10)

Depending on the problem size and the available resourcesfor experimentation, enough experiments may be performed tobuild a model that include two-factor interactions. Theexperiment results are used to determine the factor significanceon the system responses and to eliminate those which areunimportant. Another task in a sequential strategy is to find theregion of interest for fitt ing a higher order response surfacemodel (Bernardo, et al., 1992; Montgomery, 1991). In thetraditional response surface approach it is fairly simple to use asteepest descent method to find this region of interest.However, in a multiobjective design problem, where more thanone goal and several constraints are present this task is ratherdifficult. Heuristic rules may be applied to reduce the designspace (Chen, 1996). Goals and constraints for each experimenthave to be evaluated. In this study, we also use the simplemodel of Eq. (10) within a compromise DSP to find thedirection of good designs.

After screening, result evaluation and design problemreduction (number of variables, bounds), second-order responsesurface models are created to replace the original strcuturalanalysis with an efficient, usually polynomial approximation.

These models include two-factor interactions and quadratic termsto account for nonlinearity in the model. Central compositedesigns are probably the most widely used designs to fitsecond-order response surface models and studying second-ordereffects. A CCD generally consists of a two-level full orfractional factorial design portion, where the two levels arecoded by -1 and +1, two star points on the axis of each factor ata distance of - and + from the center, and one or more centerpoints. A useful property of a CCD is that it may be built upfrom first-order designs, as used for screening, by adding the starpoints and center points. From the experiment results, second-order models of the form

y = b 0 + bii

k

x i + biiik

x i2 + bij jii< j x i x j (11)

are fitted by means of regression analysis based on a least-squaremethod. It is necessary to determine the adequacy of the leastsquares fit since the true functional relationship is usuallyunknown. It is helpful to examine a normal probability plotand a plot of residuals vs. fitted values (Box, et al., 1978) .

Computer experiments differ from physical experiments in thatthere is no random error (Sacks, et al., 1989; Welch, et al.,1990). Therefore the adequacy of a response surface model fittedto the observed data is determined solely by systematic bias,e.g., the difference between the assumed and the exact model.The classical notions of experimental unit, blocking,

replication, and randomization are no longer relevant. Such aresponse surface has to be fit for each of the state variablesdescribing the behavior of a system which is then used todescribe the system goals and constraints.

For the described experimental designs it is assumed that allparameters are independent, i.e., the levels of one parameter areindependent of the levels of all other parameters. However, inshape optimization problems this may not be the case and theresulting designs will have completely different factor levels.For the shape of a B EZIER curve as shown in Figure 2, the startand end control point coordinates can be selected orbitrarily,while coordinates of the intermediate control points are definedthrough some kind of functional relationship. If factordependencies are present it is necessary to determine the absolutebounds for the dependent variables. This is a problem dependenttask and based on the functional relationship.

The question to be answered is how does the factordependency affect the experimental designs as described before?The advantageous property of orthogonality will be lost. Infirst-order designs, orthogonality is the optimal design propertyas it minimizes the variances of the regression coefficients, i.e.,a greater precision of estimates is obtained from orthogonaldesigns. In linear regression analysis the assumed model can bewritten in matrix form as

y X= + , (12

where y is the vector of responses, X is the vector of theregressor variable levels, is the vector of the regressioncoefficients, and is the vector of random errors. For findingthe least-square estimates, the inverse matrix ( X TX )-1 has to befound. If experimental designs are orthogonal, ( X TX )-1 idiagonal and simple to obtain computationally. Furthermore,the estimates of all the regression coefficients are uncorrelatedand the covariance is zero.

x1

x2

-1

1

0.33

- 0.33

x1

x2

-1

1

Figure 3 -- Factorial design and CCD with parameterdependency


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Factorial designs are orthogonal and CCDs can be madeorthogonal through the number of center points. Unfortunately,if parameter dependencies are present, the designs will be non-orthogonal and correlation between regression coefficients willexist. The variances for the regression coefficients will bedifferent (Box, et al., 1978; Box and Draper, 1987).

In Figure 3, a two factor factorial and a CCD are consideredwhere the levels of factor x 2 depends on the levels of factor x 1 .Factor x 1 has two levels in the factorial design, while factor x 2has four levels. It is not necessarily the case that the differencebetween the two x 2 levels is the same for each level of x 1 . In aCCD, factor x 1 has five levels, while factor x 2 has nine levels,one for each experiment. While standard designs can be createdinitially, the now relative 1 and levels of the dependentfactor x 2 need to be transformed into their new levels. Then,they have to be normalized again to absolute values between -1and +1. The factor values of Figure 3 are given in Table 2.

Table 2 -- Non-orthogon al experimental designs

2 2-Factorial CC DE x p . x 1 x 2 E x p . x 1 x 2

1 -1 -1 1 -1 -12 -1 0.33 2 -1 0.333 1 -0.33 3 1 -0.334 1 1 4 1 1

5 -1.41 -0.476 1.41 0.477 0 -0.948 0 0.949 0 0

The shape optimization problem of a flywheel serves as anexample to illustrate the presented approach, where non-orthogonal, low-order screening experiments are used to reducethe design problem and non-orthogonal, second-order responsesurface models are used for efficiently modeling the trade-off between two conflicting design objectives.

5 SOLUTION TO THE DESIGN PROBLEMIn this section, we illustrate our screening strategy to gain

knowledge about the structural behavior and our approach tomodel the trade-off among two design objectives efficientlythrough employing response surfaces during designoptimization. We use non-orthogonal experimental design asintroduced in Section 4 for this purpose.

5. 1 Step 1: Screening and Model ReductionIn the current model (Figure 2), the r-coordinates of control

points P 2 , P 3 can vary within specified bounds relative to P 1

and P 4 . Their relative lower and upper bounds are -1 and +1respectively. The normalized values P 2r

* and P 3r* are then

converted into their real values according to some specification.In this model there are eight design variables x , which arerepresented by geometrical dimensions and control pointcoordinates. A two-level factorial design with 16 experiments

plus one experiment for the center point is selected for screening(Step 1, see Section 1.3). For this design, the main effects areonly confounded with higher order interactions but not withtwo-factor interactions. The analysis results are shown inTable 3, where special focus has to be on the normalized levelsfor P 2r and P 3r. Values for the disk thickness are not consideredin this evaluation, since they are obtained independently fromthe geometry definition.

The results for the mass, the kinetic energy and themaximum stress of the disk indicate a large range for each of theresponses (state variables). The statistical analysis is doneusing a model for the main effects only. Various calculations,such as the R 2-value, analysis of variance, a table of parameterestimates as well as visual graphs like the prediction profile andPareto plot for scaled parameter estimates provide an indicationof the significance of each factor. The value R 2 measures theproportion of the variation around the mean explained by themodel . If R 2 is 1 the model fits perfect. The effects have to beevaluated for each of the responses. For this model, consideringthe main effects only, the R 2-value ranges between 0.82 and0.86. The fit for mass is the best (0.86), the fit for stress is theworst (0.82). The commercial software package JMP (N.N.,1995) is used to perform various statistical analyses.

In Figure 4, the prediction profiles for the three disk responses are displayed. The low and high values for each factor

are shown on the horizontal axis and for each response on thevertical axis. The lines and markers within the plots show theprediction profile and the 95% confidence interval for thepredicted values is shown by error bars above and below eachmarker. We easily see that r o has the largest effect on all threeresponses. An increase in r o will result in an increase of allresponses. A stress reduction is indicated for an increase of hand P 2z , while the effect is also positive (increase) on thekinetic energy. Horizontal prediction profiles indicate a loweffect of a factor on that response, e.g., r i for all three responses.

The evaluation of the screening experiments is supportedwith a so-called Pareto plot of the scaled parameter estimates.Each parameter estimate is scaled through its division by the

sum of all estimates. In a Pareto plot the parameters are orderedby their importance, i.e., the smaller the scaled estimate, theless its significance on the response. The values are scaled, notnormalized estimates. This means they might be correlated andcan have unequal variances (N.N., 1995). The line starting atthe first factor represents the sum of the estimates. It ends at1.0. The scaled parameter estimates for the kinetic energy areshown in Figure 5. Radius r o is the most important factor witha vlaue greater than 0.5, thus, accounting for more than half the


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variation. The parameters P 2r, P 3r, h i, and r i have values of lessthan 0.05, thus their effect is very small. The line representsthe sum of all estimates in the shown order.

0.2 0.4 0.6 0.8rohoP 3zP 2zP 2rP 3rhiri

0.0 1.0scaled estimate value

Figure 5 -- Scaled estimates for kinetic energy

The significance of factors r i and P 2r is very low for allresponses. These factors will be dropped from the designmodel, i.e., they will be set to constant values (Step 2). We areaware that interactions have not been taken into account, sothere is some risk involved in dropping these factors. It is amore difficult task to reduce the bounds on the remaining factors

to narrow down the design space for developing responsesurfaces. We use Eq. (10) within a compromise DSP to find adirection for satisficing designs. With maximizing the kineticenergy at highest priority, we obtain a design where more massis accumulated near the inner and outer rim, while the interior isvery thin. The outer radius r o is close to the upper bound,while the control point coordinates P 2z and P 3z reach their lowerbound. Verifying the design point with the original model, we

Table 3 -- Experimental design results

Exp. r i r o h i h o P 2z P 3z P 2r P 3r m (kg) T (kJ) r (MPa)

1 -1 -1 -1 -1 -1 -1 -1 -1 2.5096 3.0424 24.0272 -1 -1 -1 1 1 1 -0.648 -1 18.444 21.437 41.5393 -1 -1 1 -1 1 1 -1 -0.751 15.107 13.113 19.479

4 -1 -1 1 1 -1 -1 -0.648 -0.751 11.127 11.966 18.9575 -1 1 -1 -1 1 -1 0.772 0.943 64.721 483.35 234.186 -1 1 -1 1 -1 1 -0.461 0.943 137.29 1508.9 493.697 -1 1 1 -1 -1 1 0.772 0.072 94.171 716.89 132.708 -1 1 1 1 1 -1 -0.461 0.072 148.63 1431.9 190.319 1 -1 -1 -1 -1 1 -0.42 -0.694 5.5037 7.4222 32.306

1 0 1 -1 -1 1 1 -1 -0.64 -0.694 9.475 12.701 30.9821 1 1 -1 1 -1 1 -1 -0.42 -0.85 8.3694 8.2 18.9241 2 1 -1 1 1 -1 1 -0.64 -0.85 13.679 18.024 29.2951 3 1 1 -1 -1 1 1 -0.102 0.222 103.73 835.3 242.221 4 1 1 -1 1 -1 -1 1 0.222 90.922 1140.1 408.471 5 1 1 1 -1 -1 -1 -0.102 1 42.515 282.28 90.2531 6 1 1 1 1 1 1 1 1 192.82 1779.7 223.31

1 7 0 0 0 0 0 0 -0.187 -0.132 47.813 200.74 98.929

ri ro hohi P2z P 3z P2r P 3r

r

T

m

192.8222

2.5096

5 9 . 2 2 5 6 9

1779.739

3.0424

4 9 8 . 5 3 4 7

493.6933

18.9242

1 3 7 . 0 3 3 4

- 1 10 - 1

10 - 1

10 - 1

10 - 1

10 - 1

10 - 1

1-0.1873 - 1

1-0.1322

192. 822

2. 5096

1779. 74

3. 0424

493. 693

18. 9242

59. 2257

498. 535

137. 033

- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 11 0 1 1 1 1 1 10 1 0 0 0 0 - 0. 187 - 0. 132

Figure 4 -- Prediction profile of screening results


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identify a feasible design with a small over-estimation for allresponses. However, with stress reduction as the higherpriority, the comparison of analytical and original model resultsin large discrepancies. The solution is found in a region wherethe estimated model is invalid. Based on the good first results,we will reduce the bounds to 200mm ro 300mm and

0mm P2z ,P 3z 20mm (Step 2). With two objectives, thereare two regions of interest, thus a design space reduction is evenmore difficult.

5.2 Steps 2 and 3: Design using ResponseSurface Models

The next step (Step 3) in the design process is toapproximate the state variables kinetic energy, mass, and totalstress with second-order response surfaces (Eq. (11)) over thereduced design space. A CCD is used and the requiredexperimental design contains 45 experiments for the sixremaining factors. In this design, the value for the starpoints is chosen to be = 1.0. There are two reasons for thischoice: first, a value of 1would lead to infeasible designs,and second, in structures it can be important to include factorcombinations which represent corners of the design space, wheresome values might be more extreme. There is a total of 28regression variables to be estimated for each state variableduring regression analysis. The second-order model includeslinear, quadratic and interaction terms. Two of the createdresponse surface models are displayed in Figure 6. Fourvariables are kept constant at selected values for the plots. Forthe plot of the kinetic energy, r o and h o vary within theirbounds, for the reference stress it is r o and h i. These are thevariables with the largest effects on the response.

The model adequacy is checked with the R 2-value. For eachresponse surface model this value is above 0.99 which indicatesa very good fit. But this does not ensure high accuracy in thewhole design space since only a few points (27) are used torepresent the complete space. Comparing the original andapproximated models for some randomly selected points, weidentify deviations of up to about 10% for the disk stressresponse. Also the stress residuals vary between 25 MPa.This is the price we have to pay if we dont want to reduce thedesign space to a smaller region and perform additionalexperiments within that reduced space. In order to account forthe inaccuracies, it is important to verify the obtained design

points to see if constraints are violated.The response surface model is exercised for changing goalpriorities (Step 4). This is done through changing the twopriority levels f 1 , f 2 in the formulation of the deviation functionin the compromise DSP (see Section 3). First, the kineticenergy is to be maximized, second, the total stress is to beminimized to a certain target. Once the stress target is achieved,the kinetic energy is will be maximized on the second priority

level. The disk shape for the final design point is displayed inFigure 7.

200.0

250.0

300.0

225.0

275.0

20.0

100.0

40.060.0

80.0

0.0

1500.0

1000.0

500.0

T i n k J

ho in mmro in mm

200.0

250.0

300.0

225.0

275.0

20.0

100.0

40.060.0

80.0

0.0

400.0

300.0200.0

100.0

r

i n M P a

h i in mmro in mm

Figure 6 -- Response surfaces for kinetic energy Tand reference stress r

We see that a large portion of the disk mass is concentratednear the outer rim, a smaller portion around the shaft. Theinterior is very thin, reaching the specified lower disk thickness.The position of the control points is displayed, too. We obtainthe following values for the state variables (responses):m = 100.0 kg, T = 1140.96 kJ, and r = 250.0 MPa. A resultcomparison with the original model is given in Table 4.

r in mm

z i n m m

Figure 7 -- Optimal disk profile for maximizingkinetic energy


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The stress state for the radial, tangential and total stress ispresented in Figure 8. The original model has to be analyzed toobtain these results. A response surface model is onlydeveloped for the reference stress. The total stress is highest atthe inner radius. Due to this reason, no more mass can beplaced further outside. The stress distribution of the total stress

agrees well with those documented in (Sandgren and Ragsdell,1983; Vadde, 1995).

r in mm

i n

M P a

rr r

Figure 8 -- Stress states along disk radius

Previously on the second priority level, the goal forminimizing the total stress is now placed on the first levelwithin the compromise DSP. On the second priority level, thekinetic energy is maximized once the stress target is achieved.

The optimal disk profile for the stress target G = 100.0 MPa isdisplayed in Figure 9. The profile is similar but now there ismore mass accumulated near the inner rim, where the stressmaximum always occurs. The inner thickness h

i is at its upper

bound and so is the radius r o . Since r o is at the upper bound forboth solutions, it could be set constant to that value to furtherreduce the problem size.

z i n m m

r in mm

Figure 9 -- Optimal disk profile for minimizings t ress

A comparison of the response surface model results and theverification with the original model (Table 4) indicatesrelatively high accuracy for mass and kinetic energy

approximations. However, there is a 6 to 23 MPa difference forthe stress estimation, which is within the predicted range of residuals. A designer has to decide how much constraints can berelaxed and what differences can be tolerated.

Table 4 -- Result c omparis on

Max. T Min. rOriginal RSM Original RSM

T (kJ) 1135.22 1140.96 523.79 519.15

m (kg) 100.11 100.0 69.35 69.06

MPa 228.22 250.0 106.11 99.99

It is now easily and efficiently possible to exercise theresponse surface models and model the trade-off between the twoconflicting objectives. The goal targets for the stress are

changed from G = 100 MPa to 250 MPa. Since one objectiveis to be maximized (T) and the other to be minimized ( r), theconflict is better illustrated when 1/T is plotted instead of T(Figure 10). The previous profile solutions are displayed, too.We identify a convex behavior of the functional-efficientboundary curve. Within a wide range of stress values ( r = 160to 250 MPa), we achieve a high kinetic energy, but there is astrong change between r = 100 and 160 MPa. When moremass has to get transferred from the outside to the inside, thekinetic energy drops drastically for lower stress values.

1 / T

i n

k J

- 1

r in MPa

Response Surface Model

Simulation Model

Figure 10 -- Functional-efficient boundary: stressvs. reciprocal kinetic energy

A comparison of each result indicates over- andunderestimation of the stress, depending on the design spaceregion for the solu tion point. Despite this, the results aresufficient for the early stages and a designer gets a very goodfeeling how to model the objectives and how the trade-off lookslike.

In the presented approach it is not always obvious howmany experiments to perform, which regression models to use,


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and how to narrow down the design space. In the currentexample, even the original analysis of the flywheel onlyrequires a few seconds of computer time and therefore variousexperimental designs have been used for screening and responsesurface fitting to identify the best approach. If expensiveanalyses are required, the experiments have to be chosen

carefully. As the failure for finding the region of interest (fromscreening) for minimum stress indicates, success cannot beguaranteed with a low-order model. This is part of the trade-off between experimental efficiency and accuracy. However, theresponse surface model has a much higher accuracy and theresults agree well with the original analysis.

6 CLOSUREThe work in this paper represents an approach for the early

stages in designing complex structures when multipleobjectives require to efficiently and effectively model the trade-off between these objectives. A sequential experimental

strategy based on DOE is used to explore the design space(screening) and response surface models are used to approximatethe structural behavior as accurate as possible for a reduceddesign problem. Usually, only main factor effects areconsidered during the screening process to increaseexperimentation efficiency. Its a designers decision to selectgood experimental designs for this purpose, because of the trade-off between the number of experiments and the model accuracy.When the number of factors or responses increases to largevalues it might become more and more difficult to decide whichfactors can be dropped from the model. Response surfacemodels provide an efficient means for a designer to rapidly comeup with design solutions and model the trade-off for conflictinggoals. The presented method is very powerful for the earlydesign stages. Its main advantages are:

effective knowledge acquisition about the globalstructural behavior,

design problem reduction is facilitated, fast approach for modeling the trade-off in the early

design stages, robust design aspects might be included.

The task of experimentation and statistical analysisbecomes more difficult when parameter dependencies willchange the selected designs into non-orthogonal designs.

Parameter estimations are less accurate and correlated.Furthermore, it is practically much more difficult to normalizefactor levels and handle the relative bounds. This is not easilydone in advance when designing the experiments. Aspects of this type have to be considered in shape optimization problems,when a DOE-based approach is applied. The flywheel designprob lem is used as an application to illustrate this kind of problem. Future work includes additional validation of the

sequential design strategy as suggested by the RCEM and thedevelopment of general rules for this approach. Robust designaspects, also part of the RCEM, have to be addressed in thecurrent model to include quality considerations and the presenceof uncertainty in the model. The statistical consequences of non-orthogonal designs have to be evaluated in greater detail.

ACKNOWLEDGMENTSThis research is funded by the German Academic Exchange

Service (DAAD) with a DAAD Post Graduate Fellowshipsupported by the Second Special University Program.

REFERENCESBarthelemy, J. F. M. and Haftka, R. T., 1993,

Approximation concepts for optimum structural design - areview, Structural Optimization, Vol. 5, pp. 129-144.

Bascaran, E., Mistree, F. and Bannerot, R. B., 1987,

Compromise: An Effective Approach for Solving Multi-objective Thermal Design Problems, EngineeringOptimization, Vol. 12, No. 3, pp. 175-189.

Berger, M. and Porat, I., 1988, Optimal design of arotating disk for kinetic energy storage, Journal of Applied

Mechanics, Transactions of the ASME, Vol. 55, No. 1, pp.164-170.

Bernardo, M. C., Buck, R., Liu, L., Nazaret, W. A.,Sacks, J. and Welch, W. J., 1992, Integrated Circuit DesignOptimization Using a Sequential Strategy, IEEE Transactionson Computer-Aided Design, Vol. 11, No. 3, pp. 361-372.

Bhavikatti, S. S. and Ramakrishnan, C. V., 1980,Optimum shape design of rotating disks, Computers and

Structures, Vol. 11, pp. 397-401.Box, G., Hunter, W. and Hunter, J., 1978, Statistics for

Experimenters , Wiley, Inc., New York.Box, G. E. P. and Draper, N. R., 1987, Empirical Model-

building and Response Surfaces, John Wiley & Sons, NewYork, NY.

Chen, W., 1995, A Robust Concept Exploration Methodfor Configuring Complex Systems, Ph.D. Dissertation,Georgia Institute of Technology.

Chen, W., Allen, J. K., Mavris, D. and Mistree, F., 1996,A Concept Exploration Method for Determining Robust Top-Level Specifications, Engineering Optimization, Vol. 26, pp.137-158.

Eschenauer, H. A., Beer, R. , Lautenschlager, U. andHillmer, P., 1994, On the Modeling of a Pressure GasInsulation Component - A Multidisciplinary EngineeringTask, 5th AIAA/USAF/NASA/ISSMO Symposium on

Multidisciplinary Analysis and Optimization, Panama City,Florida, pp. 423-433.

Eschenauer, H. A., Geilen, J. and Wahl, H. J., 1993,SAPOP - An Optimization Procedure for Multicriteria


12/12


Structural Design , In: Hrnlein, H. R. E. M. and Schittkowski,K. (Eds.), 1993, Software Systems for Structural Optimization ,Birkuser Verlag, Basel, pp. 207-227.

Eschenauer, H. A., Koski, J. and Osyczka, A., 1990, Multicriteria Design Optimization - Procedures and Applications , Springer, Berlin, Heidelberg, New York.

Grudkowski, T. W., Dennis, A. J., Meyer, T. G. andWawrzonek, P. H., 1996, Flywheels for Energy Storage,SAMPE Journal, Vol. 32, No. 1, pp. 65-69.

Hrnlein, H. R. E. M. and Schittkowski, K. (Eds.), 1993,Software Systems for Structural Optimization , BirkuserVerlag, Basel.

Ignizio, J. P., 1985, Multiobjective MathematicalProgramming via the MULTIPLEX Model and Algorithm,

European Journal of Operational Research, Vol. 22, pp. 338-346.

Mistree, F., Hughes, O. F. and Bras, B. A., 1993, TheCompromise Decision Support Problem and the AdaptiveLinear Programming Algorithm, Structural Optimization:Status and Promise , AIAA, Washington, D.C., pp. 247-286.

Mistree, F., Patel, B. and Vadde, S., 1994, On ModelingMultiple Objectives and Multi-Level Decisions in ConcurrentDesign, ASME Advances in Design Automation,Minneapolis, September 11-14, pp. 151-161.

Montgomery, D., 1991, Design and Analysis of Experiments , John Wiley & Sons, New York, NY.

Mortenson, M. E., 1985, Geometric Modeling , John Wiley& Sons, New York.

N.N., 1995, JMP, Statistics and Graphics Guide , SASInstitue Inc., Cary, NC, USA.

Phadke, M. S., 1989, Quality Engineering using Robust Design , Prentice Hall, Englewood Cliffs, New Jersey.

Rao, J. R. J., Badhrinath, K., Pakala, R. and Mistree, F.,1996, A Study of Optimal Design under Conflict usingModels of Multi-Player Games, Engineering Optimization (in

press).Roux, W. J., Stander, N. and Haftka, R. T., 1996,

Response Surface Approximations for StructuralOptimization, Sixth AIAA/USAF/NASA/ISSMO Symposiumon Multidisciplinary Analysis and Optimization, Bellevue, WA,pp. 565-578.

Sacks, J., Schiller, S. B. and Welch, W. J., 1989,Designs for Computer Experiments, Technometrics, Vol. 31,No. 1, pp. 41-47.

Sandgren, E. and Ragsdell, K. M., 1983, OptimalFlywheel Design with a general Thickness FormRepresentation, Journal of Mechanism, Transmissions, and

Automation in Design, Vol. 105, pp. 425-433.Schoofs, A. J. G., Klink, M. B. M. and Van Campen, D.

H., 1992, Approximation of structural optimization problemsby means of designed numerical experiments, StructuralOptimization, Vol. 4, pp. 206-212.

Simon, H. A., 1981, The Sciences of the Artificial , TheMIT Press, Cambridge, Mass.

Vadde, S. , 1995, Modeling Multiple Objectives andMultilevel Decisions in Concurrent Design of EngineeringSystems, M.S. Thesis, Georgia Institute of Technology.

Weinert, M., 1994, Sequentialle und parallele Strategien

zur optimalen Auslegung komplexer Rotationsschalen.,Dissertation, Univerity of Siegen, Siegen, Germany.

Welch, W. J., Yu, T. K., Kang, S. M. and Wu, J., 1990,Computer Experiments for Quality Control by ParameterDesign, Quality Technology, Vol. 22, pp. 15-22.

fm.lautenschlager.confpro.dac 3961.1997

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